1. EE565 Advanced Image Processing Copyright Xin Li 2008 1 Further Improvements Gaussian scalar mixture (GSM) based denoising* (Portilla et al.’ 2003) Instead of estimating the variance, it explicitly addresses the issue of uncertainty with variance estimation Hidden Markov Model (HMM) based denoising (Romberg et al.’ 2001) Build a HMM for wavelet high-band coefficients (refer to the posted paper)

2. EE565 Advanced Image Processing Copyright Xin Li 2008 2 Gaussian Scalar Mixture (GSM) Model definition: u~N(0,1) Noisy observation model Gaussian pdf scale (variance) parameter

3. EE565 Advanced Image Processing Copyright Xin Li 2008 3 Basic Idea In spatially adaptive Wiener filtering, we estimate the variance from the data of a local window. The uncertainty with such variance estimation is ignored. In GSM model, such uncertainty is addressed through the scalar z (it determines the variance of GSM). Instead of using a single z (estimated variance), we build a probability model over z, i.e., E{x|y}=E z {E{x|y,z}}

4. EE565 Advanced Image Processing Copyright Xin Li 2008 4 Posterior Distribution where Due to is so-called Jeffery’s prior Question: What is E{x c |y,z}? Bayesian formula (proof left as exercise)

5. EE565 Advanced Image Processing Copyright Xin Li 2008 5 GSM Denoising Algorithm http://decsai.ugr.es/~javier/denoise/index.htmlMATLAB codes available at:

6. EE565 Advanced Image Processing Copyright Xin Li 2008 6 Image Examples Noisy,  =50 (MSE=2500) denoised (MSE=201)

7. EE565 Advanced Image Processing Copyright Xin Li 2008 7 Image Examples (Con’d) Noisy,  =10 (MSE=100) denoised (MSE=31.7)

8. EE565 Advanced Image Processing Copyright Xin Li 2008 8 Image Denoising Theory of linear estimation Spatial domain denoising techniques Conventional Wiener filtering Spatially adaptive Wiener filtering Wavelet domain denoising Wavelet thresholding: hard vs. soft Wavelet-domain adaptive Wiener filtering Latest advances Patch-based image denoising Learning-based image denoising

9. EE565 Advanced Image Processing Copyright Xin Li 2008 9 Similar Patches Self-similarity is a fundamental property of nature

10. EE565 Advanced Image Processing Copyright Xin Li 2008 10 Patch-based Texture Synthesis Self-similarity allows us to synthesize “new” texture patterns from a small-size sample

11. EE565 Advanced Image Processing Copyright Xin Li 2008 11 Patch-based Denoising (NL- mean) Image denoising via nonlocal mean (CVPR’2005 Best Paper Honorable Mention) Noisy patches w1w1 + wNwN w2w2 Linear combination v(N i ) v(N j )

12. EE565 Advanced Image Processing Copyright Xin Li 2008 12 Patch-based Denoising (BM3D) WD T T -1 ThresholdingWD = Noisy patches Denoised patches

13. EE565 Advanced Image Processing Copyright Xin Li 2008 13 State-of-the-art Result MSE=100 MSE=17

14. EE565 Advanced Image Processing Copyright Xin Li 2008 14 Learning-based Denoising Training Data ? cleannoisydenoised

15. EE565 Advanced Image Processing Copyright Xin Li 2008 15 Denoising Summary How improved image models improve the denoising performance Spatial to transform, complete to overcomplete, Wiener filtering to GSM There is a trend of from local to nonlocal, however, the pursuit has just started You are encouraged to take patch-based nonlocal denoisng as your final project topic Nobody can claim that he/she has solved the denoising problem

16. EE565 Advanced Image Processing Copyright Xin Li 2008 16 A Modeler’s View on Denoising Spatial-domain models Transform-domain models Stationary Gaussian Non-Stationary Gaussian Conventional Wiener filtering Stationary GGD Non-Stationary Gaussian Mixture Model Spatial-domain Spatially adaptive Wiener filtering Wavelet thresholding Wavelet-domain Spatially adaptive Wiener filtering BLS-GSM Algorithm Nonparametric (patch-based) Latest advances